Finite Group Databasebeta

← #10827334

Subgroups — #10827334

Order 768.

1690 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#547C211yesmaximal, characteristic, complemented
2C2×#5036C211yesmaximal, characteristic, complemented
3#82403C211yesmaximal, characteristic, complemented
4C22×#809C211yesmaximal, characteristic, complemented
5#71792C212yesmaximal, complemented
6#71684C212yesmaximal, complemented
7#71681C211yesmaximal, characteristic, complemented
8#71613C212yesmaximal, complemented
9#71610C211yesmaximal, characteristic, complemented
10#71560C212yesmaximal, complemented
11#71553C211yesmaximal, characteristic, complemented
12#23960—31nomaximal, Sylow
13C22×C3×#41—21no—
14C23×D12C2211yescharacteristic
15C3×#547—24no—
16C3×#547—24no—
17C2×C3×C4⋊Q8C2211yescharacteristic, complemented
18C2×C3×#123—21no—
19C2×C3×#123—21no—
20C2×C3×C22⋊Q8—21no—
21C2×C3×#121C2211yescharacteristic
22C2×C3×#121C2211yescharacteristic, complemented
23C2×C3×C4×D4C2211yescharacteristic
24C2×C3×C4×D4C2211yescharacteristic, complemented
25C22×C3×C4⋊C4C2211yescharacteristic
26C22×C3×C4⋊C4C2211yescharacteristic, complemented
27C2×#937—21no—
28C22×#213C2212yes—
29C2×#930C2212yes—
30C23×C3⋊C8C2211yescharacteristic
31C2×#832C2212yescomplemented
32#5212C2211yescharacteristic
33C2×#830C2211yescharacteristic
34C2×#830C2212yescomplemented
35#5036—24no—
36#4991—21no—
37C2×#809—24no—
38C2×#809C2212yes—
39C2×#809C2212yes—
40C2×#809C2212yes—
41#4960—21no—
42#4955—21no—
43#4939C2212yes—
44#4529C2212yescomplemented
45#4529C2212yescomplemented
46#4529C2212yescomplemented
47#4524C2212yescomplemented
48C2×#547D311yescharacteristic, complemented
49C2×#447—31no—
50C22×#102—31no—
51#2408—31no—
52#2023—32no—
53#1999—32no—
54#1893—31no—
55#1879—32no—
56#1874—32no—
57#1869—31no—
58#1823—31no—
59C24×D3—21no—
60C2×C3×#41—28no—
61C2×C3×#41—44no—
62C2×C3×#41—44no—
63C22×C3×Q8—21no—
64C22×C3×D4—21no—
65C22×C3×D4—21no—
66C22×C3×D4D411yescharacteristic
67C22×C3×D4C2311yescharacteristic
68C23×C3×C4—21no—
69C23×C3×C4—21no—
70C23×C3×C4—21no—
71C23×C3×C4C2311yescharacteristic
72C23×C3×C4C2311yescharacteristic
73C22×#67—22no—
74C22×D12—22no—
75C22×D12—24no—
76C22×D12—22no—
77C22×D12—21no—
78C22×D12—42no—
79C22×D12C2312yes—
80C3×C4⋊Q8—22no—
81C3×C4⋊Q8—22no—
82C3×#123—28no—
83C3×#123—28no—
84C3×C22⋊Q8—28no—
85C3×#121—22no—
86C3×#121—22no—
87C3×#121—22no—
88C3×#121D414yescomplemented
89C3×C4×D4—22no—
90C3×C4×D4—22no—
91C3×C4×D4—22no—
92C3×C4×D4—22no—
93C2×C3×C4⋊C4—21no—
94C2×C3×C4⋊C4—22no—
95C2×C3×C4⋊C4—21no—
96C2×C3×C4⋊C4—21no—
97C2×C3×C4⋊C4—22no—
98C2×C3×C4⋊C4—22no—
99C2×C3×C4⋊C4D411yescharacteristic
100C2×C3×C4⋊C4D414yescomplemented
101C2×C3×C4⋊C4C2311yescharacteristic
102C2×C3×C4⋊C4C2311yescharacteristic
103C2×C3×C4⋊C4C2311yescharacteristic
104C2×C3×C22⋊C4—21no—
105C2×C3×C22⋊C4—21no—
106C2×C3×C22⋊C4—21no—
107C2×C3×C22⋊C4D411yescharacteristic
108C2×C3×C22⋊C4D411yescharacteristic
109C2×C3×C42C2311yescharacteristic
110C2×#217—21no—
111#937—44no—
112C2×#213—24no—
113C2×#213—24no—
114C2×#213—42no—
115#930—24no—
116#930—42no—
117C2×(C2×D3)⋊C4—22no—
118C2×(C2×D3)⋊C4—22no—
119C2×(C2×D3)⋊C4—22no—
120C2×C4⋊Dic3C2312yes—
121C22×C3⋊C8—22no—
122C22×C3⋊C8—42no—
123C22×C3⋊C8C2312yes—
124C22×C3⋊C8C2312yes—
125#832—42no—
126C3⋊#95—21no—
127#830—22no—
128#830—42no—
129#809—24no—
130#809—24no—
131#809—24no—
132C22×#41—21no—
133C23×D4—31no—
134C22×D8—32no—
135C23×C8—31no—
136#547—24no—
137#547—24no—
138C2×C4⋊Q8C2×D311yescharacteristic, complemented
139C2×#123—21no—
140C2×#123—21no—
141C2×C22⋊Q8—21no—
142C2×#121—32no—
143C2×#121C2×D311yescharacteristic
144C2×#121C2×D311yescharacteristic, complemented
145C2×#120—61no—
146C2×C4×D4C2×D311yescharacteristic
147C2×C4×D4C2×D311yescharacteristic, complemented
148C22×C4⋊C4C2×D311yescharacteristic
149C22×C4⋊C4C2×D311yescharacteristic, complemented
150#447—64no—
151C2×#102—32no—
152C2×#102—32no—
153C2×#102—32no—
154C2×#102—32no—
155C2×#102—64no—
156C2×C22⋊C8—31no—
157C2×C22⋊C8—32no—
158#394—61no—
159#392—61no—
160#390—32no—
161#380—31no—
162#379—61no—
163#340—32no—
164#340—32no—
165#340—32no—
166#336—32no—
167C24×C3—21no—
168C24×C3C2×D411yescharacteristic
169C24×C3C2×D411yescharacteristic
170C23×D3—22no—
171C23×D3—44no—
172C23×D3—42no—
173C23×D3—42no—
174C23×D3—42no—
175C23×D3—82no—
176C3×#41—432no—
177C2×C3×Q8—24no—
178C2×C3×Q8—42no—
179C2×C3×Q8—42no—
180C2×C3×D4—22no—
181C2×C3×D4—22no—
182C2×C3×D4—24no—
183C2×C3×D4—24no—
184C2×C3×D4—21no—
185C2×C3×D4—22no—
186C2×C3×D4—22no—
187C2×C3×D4—42no—
188C2×C3×D4—42no—
189C2×C3×D4—42no—
190C2×C3×D4—42no—
191C2×C3×D4D814yescomplemented
192C2×C3×D4C2×D412yes—
193C22×C3×C4—22no—
194C22×C3×C4—22no—
195C22×C3×C4—24no—
196C22×C3×C4—24no—
197C22×C3×C4—24no—
198C22×C3×C4—21no—
199C22×C3×C4—21no—
200C22×C3×C4—21no—
201C22×C3×C4—21no—
202C22×C3×C4—21no—
203C22×C3×C4—22no—
204C22×C3×C4—22no—
205C22×C3×C4—42no—
206C22×C3×C4—42no—
207C22×C3×C4—42no—
208C22×C3×C4—42no—
209C22×C3×C4—42no—
210C22×C3×C4—42no—
211C22×C3×C4C2×D411yescharacteristic
212C22×C3×C4C2×D411yescharacteristic
213C22×C3×C4C2×D411yescharacteristic
214C22×C3×C4C2×D412yes—
215C22×C3×C4C2×D412yes—
216C22×C3×C4C2×D411yescharacteristic
217C22×C3×C4C2411yescharacteristic
218C2×#67—44no—
219C2×#67—44no—
220C2×#67—82no—
221C22×Dic3—22no—
222C2×D12—28no—
223C2×D12—22no—
224C2×D12—22no—
225C2×D12—22no—
226C2×D12—22no—
227C2×D12—22no—
228C2×D12—48no—
229C2×D12—44no—
230C2×D12—44no—
231C2×D12—42no—
232C2×D12#4114yescomplemented
233C3×C4⋊C4—24no—
234C3×C4⋊C4—24no—
235C3×C4⋊C4—24no—
236C3×C4⋊C4—22no—
237C3×C4⋊C4—21no—
238C3×C4⋊C4—21no—
239C3×C4⋊C4—21no—
240C3×C4⋊C4C2×D412yes—
241C3×C4⋊C4C2×D412yes—
242C3×C4⋊C4C2×D412yes—
243C3×C22⋊C4—24no—
244C3×C22⋊C4—24no—
245C3×C22⋊C4—22no—
246C3×C22⋊C4—24no—
247C3×C22⋊C4—22no—
248C3×C42—21no—
249C3×C42—21no—
250#217—42no—
251#213—48no—
252(C2×D3)⋊C4—44no—
253(C2×D3)⋊C4—44no—
254(C2×D3)⋊C4—82no—
255C4⋊Dic3—22no—
256C4⋊Dic3—22no—
257C2×C3⋊C8—22no—
258C2×C3⋊C8—22no—
259C2×C3⋊C8—22no—
260C2×C3⋊C8—22no—
261C2×C3⋊C8—22no—
262C2×C3⋊C8—22no—
263C2×C3⋊C8—44no—
264C25—61no—
265C2×#41—28no—
266C2×#41—44no—
267C2×#41—44no—
268C22×Q8—21no—
269C22×D4—21no—
270C22×D4—21no—
271C22×D4—32no—
272C22×D4—64no—
273C22×D4—62no—
274C22×D4—62no—
275C22×D4—62no—
276C22×D4—61no—
277C22×D4—122no—
278C22×D4#6711yescharacteristic
279C22×D4C22×D311yescharacteristic
280C23×C4—21no—
281C23×C4—21no—
282C23×C4—21no—
283C23×C4C22×D311yescharacteristic
284C23×C4C22×D311yescharacteristic
285C2×D8—64no—
286C2×D8—64no—
287C2×D8—122no—
288C22×C8—32no—
289C22×C8—32no—
290C22×C8—62no—
291C22×C8—122no—
292C4⋊Q8—22no—
293C4⋊Q8—22no—
294#123—28no—
295#123—28no—
296C22⋊Q8—28no—
297#121—22no—
298#121—22no—
299#121—22no—
300#121—64no—
301#121—122no—
302#121#6714yescomplemented
303#120—124no—
304C4×D4—22no—
305C4×D4—22no—
306C4×D4—22no—
307C4×D4—22no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—32no—
315C2×C4⋊C4#6711yescharacteristic
316C2×C4⋊C4#6714yescomplemented
317C2×C4⋊C4C22×D311yescharacteristic
318C2×C4⋊C4C22×D311yescharacteristic
319C2×C4⋊C4C22×D311yescharacteristic
320C2×C22⋊C4—21no—
321C2×C22⋊C4—21no—
322C2×C22⋊C4—21no—
323C2×C22⋊C4—61no—
324C2×C22⋊C4—62no—
325C2×C22⋊C4—62no—
326C2×C22⋊C4—62no—
327C2×C22⋊C4#6711yescharacteristic
328C2×C22⋊C4#6711yescharacteristic
329C2×C42C22×D311yescharacteristic
330#102—64no—
331#102—64no—
332#102—64no—
333#102—122no—
334C22⋊C8—62no—
335C22⋊C8—122no—
336#95—61no—
337C23×C3—22no—
338C23×C3—22no—
339C23×C3—21no—
340C23×C3—22no—
341C23×C3—21no—
342C23×C3—22no—
343C23×C3—22no—
344C23×C3—42no—
345C23×C3—41no—
346C23×C3—41no—
347C23×C3—42no—
348C23×C3#12012yes—
349C23×C3C2×D812yes—
350C23×C3C2×D812yes—
351C23×C3AGL(1,ℤ/8)12yes—
352C23×C3C22×D411yescharacteristic
353C22×D3—24no—
354C22×D3—48no—
355C22×D3—42no—
356C22×D3—42no—
357C22×D3—42no—
358C22×D3—42no—
359C22×D3—42no—
360C22×D3—42no—
361C22×D3—88no—
362C22×D3—88no—
363C22×D3—84no—
364C22×D3—84no—
365C3×Q8—48no—
366C3×D4—28no—
367C3×D4—24no—
368C3×D4—48no—
369C3×D4—48no—
370C3×D4—41no—
371C3×D4—41no—
372C2×C3×C4—21no—
373C2×C3×C4—22no—
374C2×C3×C4—21no—
375C2×C3×C4—22no—
376C2×C3×C4—22no—
377C2×C3×C4—24no—
378C2×C3×C4—21no—
379C2×C3×C4—21no—
380C2×C3×C4—22no—
381C2×C3×C4—22no—
382C2×C3×C4—22no—
383C2×C3×C4—22no—
384C2×C3×C4—22no—
385C2×C3×C4—48no—
386C2×C3×C4—48no—
387C2×C3×C4—41no—
388C2×C3×C4—41no—
389C2×C3×C4—41no—
390C2×C3×C4—41no—
391C2×C3×C4—41no—
392C2×C3×C4—41no—
393C2×C3×C4—48no—
394C2×C3×C4—41no—
395C2×C3×C4—41no—
396C2×C3×C4—42no—
397C2×C3×C4#12012yes—
398C2×C3×C4C2×D812yes—
399C2×C3×C4C22×D411yescharacteristic
400C2×C3×C4C22×D411yescharacteristic
401C2×C3×C4C2×#4112yes—
402C2×C3×C4H(5,𝔽2)12yes—
403#67—88no—
404C2×Dic3—42no—
405C2×Dic3—42no—
406C2×Dic3—42no—
407D12—24no—
408D12—24no—
409D12—48no—
410D12—42no—
411D12—42no—
412C3⋊C8—42no—
413C3⋊C8—42no—
414C24—21no—
415C24—62no—
416C24—124no—
417C24—122no—
418C24—122no—
419C24—122no—
420C24—242no—
421C24C2×#6711yescharacteristic
422C24C2×#6711yescharacteristic
423#41—432no—
424C2×Q8—24no—
425C2×Q8—42no—
426C2×Q8—42no—
427C2×D4—22no—
428C2×D4—22no—
429C2×D4—24no—
430C2×D4—24no—
431C2×D4—22no—
432C2×D4—22no—
433C2×D4—21no—
434C2×D4—34no—
435C2×D4—42no—
436C2×D4—42no—
437C2×D4—42no—
438C2×D4—42no—
439C2×D4—68no—
440C2×D4—62no—
441C2×D4—62no—
442C2×D4—62no—
443C2×D4—62no—
444C2×D4—62no—
445C2×D4—128no—
446C2×D4—124no—
447C2×D4—124no—
448C2×D4—124no—
449C2×D4—124no—
450C2×D4—122no—
451C2×D4—242no—
452C2×D4#21314yescomplemented
453C2×D4C2×#6712yes—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—62no—
473C22×C4C2×#6711yescharacteristic
474C22×C4C2×#6711yescharacteristic
475C22×C4C2×#6711yescharacteristic
476C22×C4C2×#6711yescharacteristic
477C22×C4C2×#6712yes—
478C22×C4C2×#6712yes—
479C22×C4C23×D311yescharacteristic
480D8—128no—
481C2×C8—62no—
482C2×C8—62no—
483C2×C8—62no—
484C2×C8—62no—
485C2×C8—62no—
486C2×C8—62no—
487C2×C8—124no—
488C4⋊C4—24no—
489C4⋊C4—24no—
490C4⋊C4—24no—
491C4⋊C4—22no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—62no—
496C4⋊C4—62no—
497C4⋊C4C2×#6712yes—
498C4⋊C4C2×#6712yes—
499C4⋊C4C2×#6712yes—
500C22⋊C4—24no—
501C22⋊C4—24no—
502C22⋊C4—22no—
503C22⋊C4—24no—
504C22⋊C4—22no—
505C22⋊C4—124no—
506C22⋊C4—122no—
507C22⋊C4—124no—
508C22⋊C4—242no—
509C42—21no—
510C42—21no—
511C22×C3—22no—
512C22×C3—22no—
513C22×C3—24no—
514C22×C3—24no—
515C22×C3—22no—
516C22×C3—24no—
517C22×C3—41no—
518C22×C3—41no—
519C22×C3—42no—
520C22×C3—42no—
521C22×C3—42no—
522C22×C3—42no—
523C22×C3—82no—
524C22×C3—82no—
525C22×C3#44714yes—
526C22×C3C2×#12011yescharacteristic
527C22×C3#54612yes—
528C22×C3C22×D811yescharacteristic
529C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
530C22×C3#57712yes—
531C2×D3—44no—
532C2×D3—44no—
533C2×D3—44no—
534C2×D3—816no—
535C2×D3—82no—
536C2×D3—82no—
537C2×D3—82no—
538C2×D3—82no—
539C3×C4—22no—
540C3×C4—22no—
541C3×C4—41no—
542C3×C4—42no—
543C3×C4—42no—
544C3×C4—42no—
545C3×C4—41no—
546C3×C4#52511yescharacteristic
547C3×C4#53511yescharacteristic
548C3×C4#54612yes—
549Dic3—82no—
550C23—22no—
551C23—22no—
552C23—21no—
553C23—22no—
554C23—22no—
555C23—21no—
556C23—22no—
557C23—41no—
558C23—41no—
559C23—42no—
560C23—42no—
561C23—64no—
562C23—128no—
563C23—122no—
564C23—122no—
565C23—122no—
566C23—122no—
567C23—122no—
568C23—122no—
569C23—248no—
570C23—248no—
571C23—244no—
572C23—244no—
573C23C2×#21312yes—
574C23C2×#21312yes—
575C23#93212yes—
576C23#95312yes—
577C23C22×#6711yescharacteristic
578Q8—48no—
579D4—28no—
580D4—24no—
581D4—48no—
582D4—48no—
583D4—41no—
584D4—41no—
585D4—64no—
586D4—64no—
587D4—128no—
588D4—122no—
589D4—122no—
590D4—248no—
591C2×C4—21no—
592C2×C4—22no—
593C2×C4—21no—
594C2×C4—22no—
595C2×C4—24no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—48no—
605C2×C4—48no—
606C2×C4—48no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—42no—
615C2×C4—41no—
616C2×C4—122no—
617C2×C4—122no—
618C2×C4—122no—
619C2×C4C2×#21312yes—
620C2×C4#95312yes—
621C2×C4D3×#4112yes—
622C2×C4#100912yes—
623C2×C4C22×#6711yescharacteristic
624C2×C4C22×#6711yescharacteristic
625C8—122no—
626C8—122no—
627C2×C3—24no—
628C2×C3—42no—
629C2×C3—82no—
630C2×C3#240411yescharacteristic
631C2×C3C2×#44711yescharacteristic
632C2×C3#301011yescharacteristic
633C2×C3#329412yes—
634C2×C3#330612yes—
635D3—84no—
636C22—24no—
637C22—24no—
638C22—22no—
639C22—22no—
640C22—24no—
641C22—22no—
642C22—42no—
643C22—41no—
644C22—41no—
645C22—42no—
646C22—42no—
647C22—42no—
648C22—82no—
649C22—82no—
650C22—124no—
651C22—124no—
652C22—124no—
653C22—2416no—
654C22—242no—
655C22—242no—
656C22—242no—
657C22—242no—
658C22#521714yes—
659C22C22×#21311yescharacteristic
660C22C2×#93211yescharacteristic
661C22#582912yes—
662C22#583712yes—
663C22C2×#95311yescharacteristic
664C4—22no—
665C4—22no—
666C4—42no—
667C4—42no—
668C4—42no—
669C4—41no—
670C4—41no—
671C4—242no—
672C4#582811yescharacteristic
673C4#582912yes—
674C4#583011yescharacteristic
675C3#2396011yescharacteristic, complemented, Sylow
676C2—24no—
677C2—42no—
678C2—82no—
679C2—244no—
680C2#8352512yes—
681C2#8352612yes—
682C2C2×#521711yescharacteristic
683C2#8409611yescharacteristic
684C2#8411311yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#547C2
socleC23×C3C22×D4
hypercenterC2×#547D3
hypocenterC3#23960
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#547

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×C3×#547C2semidirect
2. C2×#5036C2semidirect
3. #82403C2semidirect
4. C22×#809C2semidirect
5. #71792C2semidirect
6. #71684C2semidirect
7. #71681C2semidirect
8. #71613C2semidirect
9. #71610C2semidirect
10. #71560C2semidirect
11. #71553C2semidirect
17. C2×C3×C4⋊Q8C22semidirect
22. C2×C3×#121C22semidirect
24. C2×C3×C4×D4C22semidirect
26. C22×C3×C4⋊C4C22semidirect
31. C2×#832C22semidirect
34. C2×#830C22semidirect
44. #4529C22semidirect
45. #4529C22semidirect
46. #4529C22semidirect
47. #4524C22semidirect
48. C2×#547D3semidirect
88. C3×#121D4semidirect
100. C2×C3×C4⋊C4D4semidirect
138. C2×C4⋊Q8C2×D3semidirect
144. C2×#121C2×D3semidirect
147. C2×C4×D4C2×D3semidirect
149. C22×C4⋊C4C2×D3semidirect
191. C2×C3×D4D8semidirect
232. C2×D12#41semidirect
302. #121#67semidirect
316. C2×C4⋊C4#67semidirect
452. C2×D4#213semidirect
675. C3#23960semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×C3×#5472
C2×#50362
#824032
C22×#8092
#717922
#716842
#716812
#716132
#716102
#715602
#715532
#239603